On the Group of Homeomorphisms of the Real Line That Map the Pseudoboundary Onto Itself

http://dx.doi.org/10.4153/CJM-2006-022-8
Canad. J. Math. 58(2006), 529-547

  Published:2006-06-01
 Printed: Jun 2006

In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line $\R$, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers $\Q$ onto itself is homeomorphic to the infinite power of $\Q$ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\Q$ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to $\R^n$ for $n\ge 2$, by linking the groups in question with Erd\H os space.